Abstract
Let\(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra and Uq(\(\mathfrak{g}\)) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq(\(\mathfrak{g}\)), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq(\(\hat {\mathfrak{g}}\)) which containsV with multiplicity one and is such that all other irreducible Uq(\(\mathfrak{g}\))-components ofV have highest weight strictly smaller than the highest weight λ ofV. There is a natural partial order on the set of Uq(\(\mathfrak{g}\)) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if\(\mathfrak{g}\) is of typeA, B, C, F orG, the minimal affinization is unique up to Uq(\(\mathfrak{g}\))-isomorphism; (ii) if\(\mathfrak{g}\) is of typeD orE and λ is not orthogonal to the triple node of the Dynkin diagram of\(\mathfrak{g}\), there are either one or three minimal affinizations (depending on λ). In this paper, we show, in contrast to the regular case, that if Uq(\(\mathfrak{g}\)) is of typeD 4 and λ is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of λ.
As a by-product of our methods, we disprove a conjecture according to which, if\(\mathfrak{g}\) is of typeA n,every affinization is isomorphic to a tensor product of representations of Uq(\(\hat {\mathfrak{g}}\)) which are irreducible under Uq(\(\mathfrak{g}\)) (in an earlier paper, we proved this conjecture whenn=1).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beck, J.: Braid group action and quantum affine algebras, Preprint, MIT, 1993.
Chari, V.: Minimal affinizations of representations of quantum groups: the rank 2 case, to appear inPubl. R.I.M.S., Kyoto Univ.
Chari, V. and Pressley, A. N.: Quantum affine algebras,Comm. Math. Phys. 142 (1991), 261–83.
Chari, V. and Pressley, A. N.: Small representations of quantum affine algebras,Lett. Math. Phys. 30 (1994), 131–45.
Chari, V. and Pressley, A. N.:A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994.
Chari, V. and Pressley, A.N.: Minimal affinizations of representations of quantum groups: the simply-laced case, to appear inJ. Algebra.
Chari, V. and Pressley, A.N.: Minimal affinizations of representations of quantum groups: the non-simply-laced case,Lett. Math. Phys. 35 (1995), 99–114.
Chari, V. and Pressley, A. N.: Quantum affine algebras and their representations, Preprint, 1994.
Delius, G. W., Gould, M. D. and Zhang, Y.-Z.: On the construction of trigonometric solutions of the Yang-Baxter equation, Preprint BI-TP 94/13; UQMATH-94-02 (to appear inNuclear Phys. B).
Drinfel'd, V. G.: A new realization of Yangians and quantized affine algebras,Soviet Math. Dokl. 36 (1988), 212–6.
Lusztig, G.:Introduction to Quantum Groups, Progr. Math. 110, Birkhäuser, Boston, 1993.
Author information
Authors and Affiliations
Additional information
Both authors were partially supported by the NSF, DMS-9207701.
Rights and permissions
About this article
Cite this article
Chari, V., Pressley, A. Minimal affinizations of representations of quantum groups: the irregular case. Lett Math Phys 36, 247–266 (1996). https://doi.org/10.1007/BF00943278
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00943278